Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?

@quid: the expositions I've seen (such as kurims.kyoto-u.ac.jp/~motizuki/2010-10-abstract.pdf) are mostly teasers to make people read more. My question is about the sketch underlying the proof of the ABC conjecture, which I don't see evident there. If you have an exposition that you would recommend, I suggest that you write it as an answer.
–
James D. TaylorSep 7 '12 at 2:00

@quid: you're being stubborn. Is it not legitimate to ask questions about mathematics that is available but difficult to read and understand? @Kevin: thanks!
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James D. TaylorSep 7 '12 at 2:21

19

@James Taylor I have not made any serious attempt to read the papers. However, I can point you to two things which I think are relevant, based on hints from the introductions. The first is the very easy proof of function field ABC, which turns into an analysis of the possible branching behavior of maps $\mathbb{CP}^1 \to \mathbb{CP}^1$. For number field ABC, the source $\mathbb{CP}^1$ should turn into $\mathrm{Spec}(\mathbb{Z})$ and the target should still be $\mathbb{P}^1$ (continued).
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David SpeyerSep 7 '12 at 6:57

19

The second is that I think the Mochizuki is thinking of the target $\mathbb{P}^1$ as the $j$-line, so that maps from $\mathrm{Spec} \mathbb{Z}$ to it correspond (roughly) to elliptic curves over $\mathbb{Q}$. This is very analogous to the way that introducing an elliptic curve made FLT provable. Are these things you already understand, or would it be useful for me to write them up in more detail? Again, this is all with the caveat that I haven't looked at anything beyond the introductions, and I understand only a little bit of them.
–
David SpeyerSep 7 '12 at 7:00

7 Answers
7

I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki's claimed proof, his other work, and I make no claims about the veracity of his recent work.

First, some background which might satisfy the OP. For years, Mochizuki has been working on things related to Grothendieck's anabelian program. Here is why one might hope this is useful in attacking problems like ABC:

The main result of these papers is that a number field is determined by its absolute Galois group in the following sense: fix an algebraic closure $\bar Q / Q$, and two number fields $K$ and $L$ in $\bar Q$. Then if $\sigma: Gal(\bar Q / K) \rightarrow Gal(\bar Q / L)$ is a topological isomorphism of groups, then $\sigma$ extends to an inner automorphism $Int(\tau): g \mapsto \tau g \tau^{-1}$ of $Gal(\bar Q / Q)$. Thus $\tau$ conjugates the number field $K$ to the number field $L$, and they are isomorphic.

So while class field theory guarantees that the absolute Galois group $Gal(\bar Q / K)$ determines (the profinite completion of) the multiplicative group $K^\times$, the Neukirch-Uchida theorem guarantees that the entire field structure is determined by the profinite group structure of the Galois group. Figuring out how to recover aspects of the field structure of $K$ from the profinite group structure of $Gal(\bar Q / K)$ is a difficult corner of number theory.

Now, just as one can ask about recovering a number field from its absolute Galois group ($Gal(\bar Q / K)$ is isomorphic to $\pi_1^{et}(K)$), one can ask how much one can recover about the curve $X$ from its etale fundamental group. Any $Q$-point $x$ of $X$, i.e. map of schemes from $Spec(Q)$ to $Spec(X)$ gives a section $s_x: Gal(\bar Q / Q) \rightarrow \pi_1^{et}(X)$.

One case of the famous "section conjecture" of Grothendieck states that this gives a bijection from $X(Q)$ to the set of homomorphisms $Gal(\bar Q / Q) \rightarrow \pi_1^{et}(X)$ splitting the above exact sequence. One hopes, more generally, to recover the structure of $X$ as a curve over $Q$ from the induced outer action of $Gal(\bar Q / Q)$ on $\pi_1^{geo}(X)$. (take an element $\gamma \in Gal(\bar Q / Q)$, lift it to $\tilde \gamma \in \pi_1^{et}(X)$, and look at conjugation of the normal subgroup $\pi_1^{geo}(X)$ by $\tilde \gamma$, well-defined up to inner automorphism independently of the lift.)

As in the case of the Neukirch-Uchida theorem, there is an active and difficult corner of number theory devoted to recovering properties of rational points of (hyperbolic) curves from etale fundamental groups. Here are two dramatically difficult problems in the same spirit:

How can you describe the regulator of a number field $K$ from the structure of the profinite group $Gal(\bar Q / K)$?

Given a section $s: Gal(\bar Q / Q) \rightarrow \pi_1^{et}(X)$, how can one describe the height of the corresponding point in $X(Q)$?

I would place Mochizuki's work in this anabelian corner of number theory; I have always kept a safe and respectful distance from this corner.

Now, to say something not quite as ancient that I gleaned from flipping through Mochizuki's recent work:

Many people here on MO and elsewhere have been following research on the field with one element. It is a tempting object to seek, because analogies between number fields and function fields break down quickly when you realize there is no "base scheme" beneath $Spec(Z)$. But I see Mochizuki's work as an anabelian approach to this problem, and I'll try to describe my understanding of this below.

Consider a smooth curve $X$ over a function field $F_p(T)$. The anabelian approach suggests looking at the short exact sequence
$$1 \rightarrow \pi_1^{et}(X_{\overline{F_p(T)}}) \rightarrow \pi_1^{et}(X) \rightarrow Gal(\overline{F_p(T)} / F_p(T)) \rightarrow 1.$$
But much more profitable is to look instead at $X$ as a surface over $F_p$ which corresponds in the anabelian perspective to studying
$$1 \rightarrow \pi_1^{et}(X_{\bar F_p}) \rightarrow \pi_1^{et}(X) \rightarrow Gal(\bar F_p / F_p) \rightarrow 1.$$
But this is pretty close to looking at $\pi_1^{et}(X)$ by itself; there's just a little profinite $\hat Z$ quotient floating around, but this can be characterized (I think) group theoretically within the study of $\pi_1^{et}(X)$ itself.

I would understand (after reading Mochizuki) that looking at curves $X$ over function fields $F_p(T)$ as surfaces over $F_p$ is like looking at only the etale fundamental group $\pi_1^{et}(X)$ without worrying about the map to $Gal(\overline{F_p(T)} / F_p(T))$.

So, the natural number field analogue would be the following. Consider a smooth curve $X$ over $Q$. In fact, let's make $X = E - \{ 0 \}$ be a once-punctured elliptic curve over $Q$. Then the absolute anabelian geometry suggests that to study $X$, it should be profitable to study the etale fundamental group $\pi_1^{et}(X)$ all by itself as a profinite group. This is the anabelian analogue of what others might call "studying (a $Z$-model of) $X$ as a surface over the field with one element".

Without understanding any of the proofs in Mochizuki, I think that his work arises from this absolute anabelian perspective of understanding the arithmetic of once-punctured elliptic curves over $Q$ from their etale fundamental groups. The ABC conjecture is equivalent to Szpiro's conjecture which is a conjecture about the arithmetic of elliptic curves over $Q$.

Now here is a suggestion for number theorists who, like myself, have unfortunately ignored this anabelian corner. Let's try to read the papers of Neukirch and/or Uchida to get a start, and let's try to understand Minhyong Kim's work on Siegel's Theorem ("The motivic fundamental group of $P^1 \backslash ( 0, 1, \infty )$ and the theorem of Siegel," Invent. Math. 161 (2005), no. 3, 629–656.)

It would be wonderful if, while we're waiting for the experts to weight in on Mochizuki's work, we took some time to revisit some great results in the anabelian program. If anyone wants to start a reading group / discussion blog on these papers, I would enjoy attending and discussing.

Dear Marty, Even though I'm grateful for the reference to my paper, it is a bit outdated. For a more recent (still incomplete) view of what my intentions really are, I would recommend the introduction to this paper: front.math.ucdavis.edu/1209.0640
–
Minhyong KimSep 8 '12 at 4:07

7

By the way, I was preparing some kind of an answer to this question when I noticed it was closed. If you folks end up opening it again, maybe someone can let me know by email. My primitive knowledge of the internet hasn't extended to automatic notifications and the like.
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Minhyong KimSep 8 '12 at 4:09

Thank you for the very recent reference! I have voted to reopen, and if it reopens I will send an email.
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MartySep 8 '12 at 4:18

would you please elaborate on what you mean by abc like problems? can you tell problems of similiar flavour
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KoushikOct 2 '13 at 13:36

I would have preferred not to comment seriously on Mochizuki's work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest in this subject, especially from young people. It would obviously be very nice if they were to engage with this circle of ideas, regardless of the eventual status of the main result of interest. That is to say, the current sense of urgency to understand something seems generally a good thing. So I thought I'd give the flimsiest bit of introduction imaginable at this stage. On the other hand, as with many of my answers, there's the danger I'm just regurgitating common knowlege in a long-winded fashion, in which case, I apologize.

For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' [It has been noted here and there that the 'Survey of Hodge Arakelov Theory I,II' papers might be reasonable alternatives.][I've just examined them again, and they really might be the better way to begin.] These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is
to decipher just a little bit those few paragraphs.

The beginning of the investigation is indeed the function field case (over $\mathbb{C}$, for simplicity), where one is given a family
$$f:E \rightarrow B$$
of elliptic curves over a compact base, best assumed to be semi-stable and non-isotrivial.
There is an exact sequence
$$0\rightarrow \omega_E \rightarrow H^1_{DR}(E) \rightarrow H^1(O_E)\rightarrow0,$$
which is moved by the logarithmic Gauss-Manin connection of the family.
(I hope I will be forgiven for using standard and non-optimal notation
without explanation in this note.) That is to say, if $S\subset B$ is the finite set of images of the bad fibers, there is a log connection
$$H^1_{DR}(E) \rightarrow H^1_{DR}(E) \otimes \Omega_B(S),$$
which does not preserve $\omega_E$. This fact is crucial, since it leads to an
$O_B$-linear Kodaira-Spencer map $$KS:\omega \rightarrow H^1(O_E)\otimes \Omega_B(S),$$ and thence
to a non-trivial map
$$\omega_E^2\rightarrow \Omega_B(S).$$
From this, one easily deduces Szpiro's inequality:
$$\deg (\omega_E) \leq (1/2)( 2g_B-2+|S|).$$
At the most simple-minded level, one could say that Mochizuki's programme has been concerned with
replicating this argument over a number field $F$. Since it has to do with differentiation on $B$, which eventually turns into $O_F$, some philosophical connection to $\mathbb{F}_1$-theory
begins to appear. I will carry on using the same notation as above, except now $B=Spec(O_F)$.

A large part of HAT is exactly concerned with the set-up necessary to implement this idea, where, roughly speaking, the Galois action has to play the role of the GM connection.
Obviously, $G_F$ doesn't act on $H^1_{DR}(E)$. But it does act on $H^1_{et}(\bar{E})$ with
various coefficients. The comparison between these two structures is the subject
of $p$-adic Hodge theory, which sadly works only over local fields rather than a global one. But Mochizuki noted long ago that something like $p$-adic Hodge theory should be a key ingredient because over $\mathbb{C}$, the comparison isomorphism
$$H^1_{DR}(E)\simeq H^1(E(\mathbb{C}), \mathbb{Z})\otimes_{\mathbb{Z}} O_B$$
allows us to completely recover the GM connection by the condition that
the topological cohomology generates the flat sections.

In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dagger}\rightarrow E,$$
(the moduli space of line bundles with flat connection on $E$)
whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the \'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form:
$$\Xi: A_{DR}=\Gamma(E^{\dagger}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$
(I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by
a calligraphic $H$. I can't seem to write calligraphic characters here.)
Here, $L$ is a suitably chosen line bundle of degree $\ell$ on $E$,
which can then be pulled back
to $E^{\dagger}$.
The inequality refers to the polynomial degree in the fiber direction of
$E^{\dagger} \rightarrow E$. The isomorphism is effected via evaluation of sections at
$$E^{\dagger}[\ell]\simeq E[\ell].$$
Finally, $$ L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete \'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.

Now for some corrections: As stated, the isomorphism is not true, and must be modified at the places of bad reduction, the places dividing $\ell$, and the infinite places.
This correction takes up a substantial portion of the HAT paper. That is, the isomorphism is generically true over $B$, but to make it true everywhere, the integral structures must be modified in subtle and highly interesting ways, while one must consider also a comparison of metrics, since these will obviously figure in an arithmetic analogue of Szpiro's conjecture. The correction at the finite bad places can be interpreted via coordinates near infinity on the moduli stack of elliptic curves as the subtle phenomenon that Mochizuki refers to as 'Gaussian poles' (in the coordinate $q$). Since this is a superficial introduction, suffice it to say for now that these Gaussian poles end up being a major obstruction in this portion of Mochizuki's theory.

In spite of this, it is worthwhile giving at least a small flavor of Mochizuki's Galois-theoretic KS map. The point is that $A_{DR}$ has a Hodge filtration defined by

$F^rA_{DR}= \Gamma(E^{\dagger}, L)^{ < r} $

(the direction is unconventional), and
this is moved around by the Galois action induced
by the comparison isomorphism. So one gets thereby a map
$$G_F\rightarrow Fil (A_{DR})$$
into some space of filtrations on $A_{DR}$.
This is, in essence, the Galois-theoretic KS map. That, is if we consider the equivalence over $\mathbb{C}$ of $\pi_1$-actions
and connections, the usual KS map measures the extent to which the GM connection moves around the Hodge filtration. Here, we are measuring the same kind of motion for the $G_F$-action.

This is already very nice, but now comes a very important variant, essential for understanding the motivation behind the IUTT papers. In the paper GTKS, Mochizuki modified this map, producing instead a 'Lagrangian' version. That is, he assumed the existence of a Lagrangian Galois-stable subspace $G^{\mu}\subset E[l]$ giving rise to another isomorphism
$$\Xi^{Lag}:A_{DR}^{H}\simeq L\otimes O_{G^{\mu}},$$
where $H$ is a Lagrangian complement to $G^{\mu}$, which I believe does not itself need to
be Galois stable. $H$ is acting on the space of sections, again via Mumford's theory.
This can be used to get another KS morphism to filtrations on $A_{DR}^{H}$. But the key point is that

$\Xi^{Lag}$, in contrast to $\Xi$, is free of the Gaussian poles

via an argument I can't quite remember (If I ever knew).

At this point, it might be reasonable to see if $\Xi^{Lag}$ contributes towards a version
of Szpiro's inequality (after much work and interpretation), except for one small problem. A subspace like $G^{\mu}$ has no
reason to exist in general.
This is why GTKS is mostly about the universal elliptic curve over a formal completion near $\infty$ on the moduli stack of elliptic curves, where such a space does exists.
What Mochizuki explains on IUTT page 10 is exactly that
the scheme-theoretic motivation for IUG was to enable the move to a single elliptic curve over $B=Spec(O_F)$, via the intermediate case of an elliptic curve 'in general position'.

To repeat:

A good 'nonsingular' theory of the KS map over number fields requires a global Galois
invariant Lagrangian subspace $G^{\mu}\subset E[l]$.

One naive thought might just be to change base to the field generated by the $\ell$-torsion, except one would then lose the Galois action one was hoping to use. (Remember that Szpiro's inequality is supposed to come from moving the Hodge filtration inside De Rham cohomology.) On the other hand, such a subspace does often exist locally, for example, at a place of bad reduction. So one might ask if there is a way to globally extend such local subspaces.

It seems to me that this is one of the key things going on in the IUTT papers I-IV.
As he say in loc. cit. he works with various categories of collections of local objects that simulate global objects. It is crucial in this process that many of the usual
scheme-theoretic objects, local or global, are encoded as suitable categories with a rich and precise combinatorial structure.
The details here get very complicated, the encoding of a scheme into
an associated Galois category of finite \'etale covers being merely
the trivial case. For example, when one would like to encode the
Archimedean data coming from an arithmetic scheme (which again, will clearly be
necessary for Szpiro's conjecture), the attempt to come up with a category of
about the same order of complexity as a Galois category gives rise to the
notion of a Frobenioid. Since these play quite a central role in Mochizuki's theory,
I will quote briefly from his first Frobenioid paper:

'Frobenioids provide a single framework [cf. the notion of a "Galois category";
the role of monoids in log geometry] that allows one to capture the essential aspects of
both the Galois and the divisor theory of number fields, on the one hand, and function
fields, on the other, in such a way that one may continue to work with, for instance,
global degrees of arithmetic line bundles on a number field, but which also exhibits the new
phenomenon [not present in the classical theory of number fields] of a "Frobenius
endomorphism" of the Frobenioid associated to a number field.'

I believe the Frobenioid associated to a number field is something close to the
finite \'etale covers of $Spec(O_F)$ (equipped with some log structure) together with metrized line bundles on them, although it's
probably more complicated. The Frobenious endomorphism for a prime $p$ is then something like
the functor that just raises line bundles to the $p$-th power.
This is a functor that would come from a map of schemes if we were
working in characteristic $p$, but obviously not in characteristic zero.
But this is part of the reason to start encoding in categories:

We get more morphisms and equivalences.

Some of you will notice at this point the analogy to
developments in algebraic geometry where varieties are encoded in categories,
such as the derived category of coherent sheaves. There as well, one has reconstruction
theorems of the Orlov type, as well as the phenomenon of non-geometric morphisms
of the categories (say actions of braid groups). Non-geometric morphisms
appear to be very important in Mochizuki's theory, such as the Frobenius above,
which allows us to simulate characteristic $p$ geometry in characteristic
zero. Another important illustrative example is a
non-geometric isomorphism between Galois groups of local fields (which can't exist
for global fields because of the Neukirch-Uchida theorem).
In fact, I think Mochizuki was rather fond of Ihara's comment that the positive
proof of the anabelian conjecture was somewhat of a disappointment, since
it destroys the possibility that encoding curves into their fundamental
groups will give rise to a richer category. Anyways, I believe the importance
of non-geometric maps of categories encoding rather conventional objects
is that

they allow us to glue together several standard
categories in nonstandard ways.

Obviously, to play this game well,
some things need to be encoded in rigid ways, while others should
have more flexible encodings.

For a very simple example that gives just a bare glimpse of the general theory, you might consider a category of
pairs $$(G,F),$$ where $G$ is a profinite topological group
of a certain type and $F$ is a filtration on $G$.
It's possible to write down explicit conditions that ensure that
$G$ is the Galois group of a local field and $F$ is its ramification filtration
in the upper numbering (actually, now I think about it, I'm not sure about 'explicit conditions' for the filtration part, but anyways). Furthermore, it is a theorem of Mochizuki
and Abrashkin that the functor that takes a local field to the corresponding
pair is fully faithful. So now, you can consider triples
$$(G,F_1, F_2),$$
where $G$ is a group and the $F_i$ are two filtrations of the right type.
If $F_1=F_2$, then this 'is' just a local field. But now you can have
objects with $F_1\neq F_2$, that correspond to strange amalgams of
two local fields.

As another example, one might take
a usual global object, such as $$ (E, O_F, E[l], V)$$ (where $V$
denotes a collection of valuations of $F(E[l])$ that restrict bijectively to
the valuations $V_0$ of $F$), and associate to it a collection of local categories
indexed by $V_0$ (something like Frobenioids corresponding to the $E_v$ for $v\in V_0$). One can then try to glue them together
in non-standard ways along sub-categories, after performing a number of non-standard transformations. My rough impression at the moment is that
the 'Hodge theatres' arise in this fashion. [This is undoubtedly a gross oversimplification, which I will correct
in later amendments.] You might further imagine that some
construction of this sort will eventually retain the data necessary to get the height of
$E$, but also have data corresponding to the $G^{\mu}$, necessary for the Lagrangian KS map.
In any case, I hope you can appreciate that a good deal of 'dismantling' and 'reconstructing,' what Mochizuki calls surgery, will be necessary.

I can't emphasize enough times that much of what I write is based on
faulty memory and guesswork. At best, it is superficial, while at worst,
it is (not even) wrong. [In particular, I am no longer sure that the GTKS map is used in an entirely direct fashion.]
I have not yet done anything with the current papers than give them a cursory glance.
If I figure out more in the coming weeks, I will make corrections.
But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.

Allow me to make one remark about set theory, about which I know next to nothing.
Even with more straightforward papers in arithmetic geometry, the question sometimes arises about Grothendieck's universe axiom, mostly because universes appear to be used in SGA4. Usually, number-theorists (like me) neither understand, nor care about such foundational matters, and questions about them are normally
met with a shrug. The conventional wisdom of course is that any of the usual
theorems and proofs involving Grothendieck cohomology theories or topoi do
not actually rely on the existence of universes, except general laziness allows us
to insert some reference that eventually follows a trail back to SGA4.
However, this doesn't seem to be the case with
Mochizuki's paper. That is, universes and
interactions between them seem to be important actors rather than conveniences.
How this is really brought about, and whether more than the universe axiom is necessary for the arguments, I really don't understand enough yet to say.
In any case, for a number-theorist or an algebraic geometer, I would guess it's still prudent to acquire a reasonable feel for the
'usual' background and motivation (that is, HAT, GTKS, and anabelian things) before worrying too much about deeper issues of set theory.

Such a great answer proves that the question was worth asking.
–
OlivierSep 8 '12 at 12:47

33

@Olivier: while I agree with your comment regarding the answer, I woould like to add that your inference regarding this particular question to me comes close to saying: the great and heroic work of the fire-fighters prove that it was worth setting the house on fire.
–
quidSep 8 '12 at 13:18

128

On MathOverflow, though, it is possible to edit the original house so that it is no longer inflammatory, while still recording the work of the firefighter.
–
Terry TaoSep 8 '12 at 18:15

6

Dear Minyong, it has been stated by experts, here on MO and elsewhere, that universes are not necessary for the uses to which EGA/SGA are put. Indeed Colin McLarty has shown that ZFC is far stronger than is necessary, with the results provable in some higher order version of arithmetic. As far as the interuniversal aspect of IUTT goes, I expect it to have a topos-theoretic interpretation. This to my mind makes switching between models of set theory cleaner i.e. functorial.
–
David RobertsSep 9 '12 at 1:27

Last revision: 10/20. (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. )

Completely rewritten. (9/26)

It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-IV could hold.

Here is an infinite set of counterexamples, assuming for convenience two standard conjectures (the first being in fact a consequence of ABC), that contradict Thm. 1.10 very badly.

Assumptions:

A (Consequence of ABC) For all but finitely many elliptic curves over $\mathbb{Q}$, the conductor $N$ and the minimal discriminant $\Delta$ satisfy $\log{|\Delta|} < (\log{N})^2$.

B (Uniform Serre Open Image conjecture) For each $d \in \mathbb{N}$, there is a constant $c(d) < \infty$ such that for every number field $F/\mathbb{Q}$ with $[F:\mathbb{Q}] \leq d$, and every
non-CM elliptic curve $E$ over $F$, and every prime $\ell \geq c(d)$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2(\mathbb{Z}/{\ell})$. (In fact, it is sufficient to take the weaker version in which $F$ is held fixed. )

Further, as far as I can tell from the proof of Theorem 1.10 of IUTTIV, the only reason for taking $F := F_{\mathrm{tpd}}\big( \sqrt{-1}, E_{F_{\mathrm{tpd}}}[3\cdot 5]
\big)$ --- rather than simply $F := F_{\mathrm{tpd}}(\sqrt{-1})$ --- was to ensure that $E$ has semistable reduction over $F$. Since I will only work in what follows with semistable elliptic curves over $\mathbb{Q}$, I will assume, for a mild technical convenience in the examples below, that for elliptic curves already semistable over $F_{\mathrm{tpd}}$, we may actually take $F := F_{\mathrm{tpd}}(\sqrt{-1})$ in Theorem 1.10.

The infinite set of counterexamples. They come from Masser's paper [Masser: Note on a conjecture of Szpiro, Asterisque 1990], as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over $\mathbb{Q}$ whose conductor $N$ and minimal discriminant $\Delta$ satisfy
$$
(1) \hspace{3cm} \frac{1}{6}\log{|\Delta|} \geq \log{N} + \frac{\sqrt{\log{N}}}{\log{\log{N}}}.
$$
(Thus, $N$ in these examples may be taken arbitrarily large. ) By (A) above, taking $N$ big enough will ensure that
$$
(2) \hspace{3cm} \log{|\Delta|} < (\log{N})^2.
$$
Next, the sum of the logarithms of the primes in the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ is $2(\log{N})^2 + o((\log{N})^2)$, so it is certainly $> (\log{N})^2$ for $N \gg 0$ big enough. Thus, by (2), it is easy to see that the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ contains a prime $\ell$ which divides neither $|\Delta|$ nor any of the exponents $\alpha = \mathrm{ord}_p(\Delta)$ in the prime factorization $|\Delta| = \prod p^{\alpha}$ of $|\Delta|$.

Consider now the pair $(E,\ell)$: it has $F_{\mathrm{mod}} = \mathbb{Q}$, and since $E$ has rational $2$-torsion, $F_{\mathrm{tpd}} = \mathbb{Q}$ as well. Let $F := \mathbb{Q} \big(
\sqrt{-1}\big)$. I claim that, upon taking $N$ big enough, the pair $(E_F,\ell)$ arises from an initial $\Theta$-datum as in IUTT-I, Definition 3.1. Indeed:

(b) of IUTT-I, Def. 3.1 is satisfied since by construction $E$ is semistable over $\mathbb{Q}$;

(c) of IUTT-I, Def. 3.1 is satisfied, in view of (B) above and the choice of $\ell$, as soon as $N \gg 0$ is big enough (recall that $\ell > (\log{N})^2$ by construction!), and by the observation that, for $v$ a place of $F = \mathbb{Q}(\sqrt{-1})$, the order of the $v$-adic $q$-parameter of $E$ equals $\mathrm{ord}_v (\Delta)$, which equals $\mathrm{ord}_p(\Delta)$ for $v \mid p > 2$, and $2\cdot\mathrm{ord}_2(\Delta)$ for $v \mid 2$;

while $\mathbb{V}_{\mathrm{mod}}^{\mathrm{bad}}$ consists of the primes dividing $\Delta$;

Finally, (d) of IUTT-I, Def. 3.1 is satisfied upon excluding at most four of Masser's examples $E$. (See page 37 of IUTT-IV).

Now, take $\epsilon := \big( \log{N} \big)^{-2}$ in Theorem 1.10 of IUTT-IV; this is certainly permissible for $N \gg 0$ large enough. I claim that the conclusion of Theorem 1.10 contradicts (1) as soon as $N \gg 0$ is large enough.

He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. (He estimates January 2013 to be a reasonable period). He confirms the following ["essentially"] anticipated revision of Theorem 1.10:

Let $E/\mathbb{Q}$ be a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then:
$$
\frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + O\big( \log{(1/\epsilon)} \big)
$$
$$
< \log{N} + \Big( \epsilon \log{N} + \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big) + o\Big( \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big),
$$
where $\omega(\cdot)$ denotes "number of prime factors." The second estimate comes from the prime number theorem in the form $\pi(t) = t/\log{t} + t/(\log{t})^2 + o\big( t/(\log{t})^2 \big)$, applied to $t := \epsilon^{-1}$, and is sharp if you restrict $\epsilon$ to the range $\epsilon^{-1} < (\log{N})^{\xi}$ with $\xi < 1$, as there nothing prevents $N$ from being divisible by all primes $p < (\log{N})^{\xi}$. In particular, as the Erdos-Stewart-Tijdeman-Masser construction is based on the pigeonhole principle, which cannot preclude that $N$ be divisible by all the primes $< (\log{N})^{2/3}$, the second estimate could very well be sharp in all the Masser examples. As it is easily seen that the bracketed term exceeds the range $\sqrt{\log{N}}/(\log{\log{N}})$ of Masser's examples, this has the implication that

On the other hand, if we take $\epsilon := (\log{N})^{-1}$ and assume $\omega(N_{\epsilon})$ bounded, this would yield $(1/6)\log{|\Delta|} < \log{N} + O(\log{\log{N}})$, just as before. (Thus, Mochizuki predicts that this last bound must hold for $N$ a large enough square-free integer such that the number of primes $< \log{N}$ dividing $N$ is bounded. I cannot see evidence neither for nor against this at the moment: again, the Masser and Erdos-Stewart-Tijdeman constructions are based on the pigeonhole principle, and do not seem to be able to exclude the small primes $< \log{N}$. So here we have an open problem by which one could probe Mochizuki's revised inequality. A reminder: in terms of the $abc$-triple, $\Delta$ is essentially $(abc)^2$, and $N = \mathrm{rad}(abc)$).

A side remark: note that the inverse $1/\ell$ of the prime level from the de Rham-Etale correspondence $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$ in Mochizuki's "Hodge-Arakelov theory" ultimately figures as the $\epsilon$ in the ABC conjecture.

[I have deleted the remainder of the 10/15 Addendum, since it is now obsolete after Mochizuki's revised comments. ]

This conversation is very interesting, but there seems to be a community consensus that we should avoid discussion of the validity of recently announced results. (Also, your post is not really an answer to the question that was asked.)
–
S. Carnahan♦Sep 16 '12 at 10:06

12

I agree. Let us keep on examining Mochizuki's papers, and hope that his daring but beautiful strategy (deforming, as it were, number fields!) really works. ABC is such a delicate problem: the slightest error in one part of the theory could easily ruin the whole proof (as with Miyaoka's earlier attempt). As for the asymptotic discrepancy with ABC near-misses, Mochizuki himself seems to be a bit perplexed (see the remark on p. 40); hopefully this will be straightened out.
–
Vesselin DimitrovSep 16 '12 at 16:07

13

If it is completely rewritten, then you should post it as a different answer, because people who upvoted/downvoted your post may change their mind and the relation with discussion in the comments are broken.
–
36minSep 26 '12 at 17:11

105

I think that people are working way, way too hard to define MathOverflow by what it isn't. Although I still like MO, too many babies have been thrown out with various bathwater. This posting by Dimitrov is as important as anything else that I have seen in MO lately. Yet a direct question about the validity of Mochizuki's work was closed down, and then this question was also closed down for a while. (Which may be a valid technical reason to revise an answer instead of posting another one.) What really is the point? It becomes rules just for the sake of rules.
–
Greg KuperbergSep 27 '12 at 16:09

21

Remark about mathematical communication: If Mochizuki's much-publicized papers need revision, then it would be much better for him to post to the arXiv for the sake of public commitment of versions. Preprints that announce important results shouldn't be a moving target. In fact, with his permission it could be done by proxy.
–
Greg KuperbergSep 28 '12 at 19:28

[The answer below is a response to an earlier version of the question that was rather different in certain respects. Minhyong Kim's answer gives excellent insight into ideas that Mochizuki had back in 2000 and that provide essential building blocks for the more recent work. But I still believe that it is too premature for a non-expert to seek insight into the new work, for reasons explained below, given that many top experts are presently trying to absorb the ideas Mochizuki developed back in 2000.]

This question appears to be inspired by an historical fallacy: the only "vision" of a proof of the Weil Conjectures that Grothendieck had when he began developing ideas related to his work on the problem (i.e., etale cohomology) was the one laid out in Weil's original paper. The yoga around the standard conjectures came much later.

That being said, although the new ABC developments are potentially very exciting, and it is understandable to want to "share in the excitement", for reasons specific to this situation it seems to be much too premature to ask for a sketch on MO or in a blog of Mochizuki's vision/proof with an expectation of insight into the new work. Let me try to indicate why this is the case.

As has been explained clearly by JSE elsewhere, there are plenty of top experts in arithmetic geometry who are presently struggling to get even a small handle on what is really going on in Mochizuki's papers (due entirely to the experts' lack of prior study of these ideas; Mochizuki's writing is extremely precise, detailed, thorough, and full of intuitive asides!). So the situation seems to be rather different from that of other tremendous advances in recent decades (by Perelman, Faltings, Wiles, etc.), for which the deep new work took place within a context that was already somewhat familiar to a good-sized community of experts in the field (who could then use their experience and expertise to quickly disseminate a "bird's eye view" to others of some of the key new ideas).

Because of the rather unique circumstances of this case, as just indicated, I believe that quid's initial urging of patience (if one isn't going to be directly engaged with the struggle to read the actual papers and the prior work upon which they depend) is appropriate.

But to end on a semi-positive note, let me explain why quid's mention of Mochizuki's survey papers is very apt. Some of those surveys are relatively short (e.g., less than 20 pages), and if you find them difficult to grok then you will get a real sense of the difficulties that a lot of top experts are current facing in their efforts to try to understand what Mochizuki has achieved. Please be patient! As quid has noted, in due time, as experts eventually come to acquire a genuine understanding of the overall structure of the arguments in these papers, plenty of expositions for wide dissemination of the ideas will emerge. Mochizuki has put a lot of effort into providing indications of his motivation and insights throughout his papers (which are a serious challenge even for top experts to absorb), and to respect his remarkable effort it seems best to engage with it directly (whether through reading the surveys or the main papers).

@James: OK, good to hear the clarification. But one of Mochizuki's survey papers does address exactly what you suggest you'd like to hear about at the end of your comment, though using the sophisticated language of moduli stacks (which, if you pretend are schemes, can be inspiring even if you don't know about stacks): see 1.3.1 of "A survey of the Hodge-Arakelov Theory of Elliptic Curves I". Mochizuki is an extremely good writer!! If you elide unclear technical issues and try to just digest the flavor, you can get a lot of inspiration. I am reminded about "reading the masters"... :)
–
grpSep 7 '12 at 16:51

11

Disambiguating note: grp is a login name I use on meta.mathoverflow. Much as I might like to take credit for the answer above, I did not write it. Hopefully the entity will identify themselves further so as to clarify. I do not ask for a change of user name though, either on my part or on the entity's part. Gerhard "Really, That Is Not Me" Paseman, 2012.09.07
–
Gerhard PasemanSep 7 '12 at 20:11

3

Dear grp, I'm curious as to why you wrote (in the comment thread above) that the function field case had nothing to do with, or provided not motivation for, the number field case. My impression was rather the opposite, that Mochizuki was/is trying to get around Faltings's "no go" theorem about an arithmetic KS map by radically reinterpreting the whole thing in very sophisticated non-linear, or anabelian, terms. I wonder if you can say more about what you meant? Best wishes, Matthew
–
EmertonSep 7 '12 at 22:21

3

Dear Emerton: As you know, there is a purely algebraic proof of the "classical" (not "Szpiro") formulation for F(t), and that is what I was pretty sure David Speyer was considering to present. That argument (as one finds in elementary expositions) is what I meant isn't relevant, much like the F(t)-version of FLT proved by bare-hands algebra rather than by genus; sorry for being unclear. I agree that the F(t)-case proved via a classifying map from a curve to a moduli space of elliptic curves, thereby highlighting the role of Kodaira-Spencer maps, is a crucial perspective in Mochizuki's work.
–
grpSep 7 '12 at 22:35

10

For later readers, I want to remark that this answer refers to an old version of the question.
–
Andy PutmanSep 8 '12 at 18:03

Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller theory for number fields" is certainly very beautiful, and was known to Mochizuki, along with the nature of the final estimate, already in 2000. (But let us recall, as a sane reminder of just how elusive the ABC conjecture has been, Miyaoka's flawed proof: did the idea of a Bogomolov-Miyaoka-Yau type bound involving arithmetic Chern numbers in Arakelov theory not seem equally beautiful, exciting, and promising?)

In brief, the main idea behind the IUTT-series is to construct, outside the rigid confines of algebraic geometry, a subtle object simulating a rank-1, Galois-stable quotient of $E[\ell]$. Here, $E/\mathbb{Q}$ is a (pretty much) arbitrary rational elliptic curve (and this is the main point: such a Galois-stable quotient will almost never exist!); and $\ell \geq 5$ is an auxiliary prime, generic for $E$ in a very mild sense, but otherwise free to optimize until the very final estimate. This is then applied to construct, in the non-linear discretized "Hodge-Arakelov theory," a comparison isomorphism between $(E^{\dagger}, <\ell)$ and $E[\ell]$, which is free of Gaussian poles at the bad places of $E$. For this then leads to a promising Galois-theoretic "Kodaira-Spencer map," as explained in Minhyong Kim's post, hopefully leading in the familiar way to the arithmetic Szpiro inequality for this very same elliptic curve: $\log{|\Delta_{\mathrm{min}}(E)|} \leq (6+\varepsilon) \log{N_E} + O_{\varepsilon}(1)$.

Let me, however, disagree with one point from M. Kim's post. My impression is that what Mochizuki calls an "initial $\Theta$-datum" - and which is, essentially, the pair of the rational elliptic curve $E$ (or equivalently, the $abc$-triple from the ABC-conjecture!) and, until the very final estimate in Ch. 2 of the fourth paper, the prime level $\ell$ - are fixed for good throughout the entire series of IUTT-papers. The deformation flavor of "Teichmuller theory" refers to dismantling the underlying number field, and not to the elliptic curve enhancement (indeed, in Mochizuki's dictionary with his own $p$-adic Teichmuller theory, it is the number field that corresponds to a hyperbolic curve; the elliptic curve enhancement corresponds to an "indigenous bundle" over the hyperbolic curve, and invites the anabelian philosophy via the \'etale fundamental group of the once-punctured elliptic curve). All the "Hodge theaters" associated to the initial $\Theta$-datum are isomorphic to one another, and form a vastly complicated $2$-dimensional non-commutative array - the "$\mathfrak{log}-\Theta$ lattice" - of non-ring theoretic translations between one another. What Mochizuki writes on p. 10 of IUTT-I is that the theory of $\Theta$-Hodge theaters "may be regarded as a sort of solution to the problem of constructing the global quotient $E[\ell] \twoheadrightarrow Q$" [needed for the application to arithmetic Kodaira-Spencer]. He does not seem to suggest that this is done by "moving the initial $E$ to a single elliptic curve via the intermediate case of an elliptic curve in general position," as M. Kim writes. (The term "elliptic curves in general position" indeed figures in Mochizuki's fourth paper, but it has a different, not-so-essential significance that comes through his entirely self-contained paper [GenEll], and whose purely technical purpose is to reduce the general ABC conjecture to the restricted version of Szpiro's inequality for $E$, in Thm. 1.10 of IUTT-IV, coming from the estimate in IUTT-III).

In particular, in sharp contrast to the Thue-Siegel-Roth tradition of Diophantine approximations, Mochizuki's program does not seem to compare different elliptic curves / $abc$-triples, all the way through to the key estimate
$$
(*) \hspace{3cm} \log{|\Delta_{\mathrm{min}}(E)|} \leq \big(6 + \varepsilon + 200/\ell\big)\log{N_E} + 12\log(\ell\varepsilon^{-7})
$$
of IUTT-IV [asserted for all primes $\ell \geq 5$ that are generic for $E$ in a rather mild sense: essentially, $\ell$ has to be prime to the degenerate places and the $q$-parameters of $E$. Also, $\varepsilon \in (0,\epsilon_0)$ is arbitrary, with $\epsilon_0$ a numerical value. ] In this sense, Mochizuki's approach - nevermind the vast technical difficulties precipitated by the non-ring theoretic simulation of a global quotient $E[\ell] \twoheadrightarrow Q$ - is entirely direct and, consequently, effective.

So what does Mochizuki actually (claim to) prove?

Start with an $abc$-triple (co-prime rational integers with $a+b+c=0$). Since the discriminant $(\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)$ of a cubic polynomial $x^3 + \cdots$ encapsulates exactly this equation, it is a profitable, traditional idea to interpret the $abc$-datum as the giving of the rational elliptic curve $E = E_{a,b,c}$ defined by the equation $y^2 = x(x-a)(x+b)$. The (apparently weaker, but virtually as powerful) ABC conjecture $abc < K_{\varepsilon}\cdot\mathrm{rad(abc)}^{3+\varepsilon}$ then translates into Szpiro's inequality: $\log{|\Delta_{\min}(E)|} \leq (6+\varepsilon)\log{N_E} + O_{\varepsilon}(1)$ between the minimal discriminant $\Delta_{\min}(E)$ and conductor $N_E$ of $E$ (which are, essentially, $(abc)^2$ and $\mathrm{rad}(abc)$). Pick the "auxiliary prime" $\ell \geq 5$ to be generic for $E$ in the sense that, essentially: (1) $\ell \nmid abc$; (2) $\ell$ does not divide the prime exponents in $abc$; (3) for $F := \mathbb{Q}( \sqrt{-1}, E[15] )$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2 (\mathbb{Z}/\ell)$. [Conjecturally, the last condition should only exclude a finite list of primes, independent of $E$!] Then Mochizuki [IUTT-IV, Thm. 1.10] claims that (*) should hold for any $\varepsilon < \epsilon_0$.

This is the essential Diophantine estimate. Anything further than that [i.e., the deduction of the full ABC conjecture in IUTT-IV, Section 2] consists of standard, and relatively straightforward reductions [such as, e.g., the use of non-critical Belyi maps] elaborated in Mochizuki's self-contained paper [GenEll]: "Arithmetic elliptic curves in general position." Mochizuki indeed writes, in his first paper, that the auxiliary prime level $\ell \geq 5$ from the Hodge-Arakelov discretized non-linear comparison isomorphisms/correspondences $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$, will be chosen in the Diophantine application to be large, roughly on the order of the height of $E$. But this comes entirely through Theorem 3.8 in [GenEll]: there, the various non-divisibility properties are ensured by simply taking $\ell$ to exceed all the primes of bad reduction / all the $q$-parameters (also, the full Galois action is ensured unconditionally). In (*), $\ell$ could be any prime satisfying the mentioned non-divisibility conditions. (This, by the way, is what I considered highly disturbing).

My apology if I have misunderstood - and misrepresented - the points from Mochizuki's papers that I have alluded to.

For people wanting to known more details without having to read all the 500 pages: the answer provided by VD is a nice survey of the first 16 pages of IUTT-I (avoiding technicalities of how the several types of Hodge theaters are actually constructed, or what are the prime-strips). For the interested reader, the first 27 pages of IUTT-I indeed give a very good introduction. However, it is better to get used with the language of Frobenioids FIRST, otherwise the exposition can be intimidating. Unfortunately, it does not hint on the actual "source of inequality" (I mean, not beyond analogies).
–
PastenSep 17 '12 at 21:28

1

(cont.) The table in page 27 of IUTT-I gives an idea of what are the roles played by some of the main objects introduced by Mochizuki (and as VD pointed out, the hyperbolic curve "is" the number field, not the elliptic curve). Anybody can read this directly from the paper, but the only reason why I am mentioning it is the following: I was very curious about the papers (as everybody else), but the first couple pages seemed very intimidating. However, after spending some time with the papers on Frobenioids then the introduction of IUTT-I became readable after all. I hope this suggestion helps!
–
PastenSep 17 '12 at 21:42

2

This sounds like sensible mathematics, far from the cries of 'no one has any idea what he's doing!' reported in the press. Creating new mathematical objects that are formal (co)limits of existing objects without the required (co)limits is a long-established tradition (e.g. negative numbers! but more recently stacks etc).
–
David RobertsSep 18 '12 at 1:01

6

Deane Yang mentioned this post on Facebook, remarking that Vesselin Dimitrov 'dared to disagree' with me. Of course Deane meant this in a good humored way. But still, I copy here my reply, in case there's any misunderstanding: Hi Deane, 'Dare to disagree' is a strong term! I still haven't had any time to really look at Mochizuki, so it's quite likely I said something stupid that was noted by Dimitrov. I'll try to look at it more carefully soon, but I hope it doesn't take much daring for people to disagree with me (in general).
–
Minhyong KimSep 18 '12 at 11:05

1

My apology for sounding argumentative in relating my impression from Mochizuki's papers: it was not my intention, and I should not have used the word "disagree."
–
Vesselin DimitrovSep 18 '12 at 20:10

I want to point out a bibliographical information that perhaps is not very well-known and can be taken as "evidence" for the possibility of applying anabelian geometry to the ABC conjecture successfully. However, I am not claiming that this is related in any sort of way to Mochizuki's work.

Here is the fact: There is a $\pi_1$ proof of the function field Szpiro conjecture (over the complex numbers, as far as I know). The proof is indeed easy and conceptually clear, you can find a nice exposition of it in some (expository) paper of Zhang, whose title is lost somewhere in my memories. (EDIT: the paper is "Geometry of algebraic points").

Anyway, I can tell you what is the key point of the argument. Let E be an elliptic fibration over the projective line L over the complex numbers. Assume that E has only multiplicative bad reduction. You can read the order of the discriminant at a point of L from the Kodaira type of the fibre, which in turn can be recovered in terms of monodromy representations of the fundamental group of L minus the points with bad fibres: smooth fibres have trivial monodromy, and the monodromy of singular fibres is determined by Dehn twists (assuming multiplicative reduction). You can look at all these local representations at once, after choosing loops to link bad points to some generic point p of L and then study the image of the global monodromy representation on the homology of the fibre above p. Choosing loops appropriately gives the usual commutator relation which in the image of the global representation gives a relation R=1 among generators of the local reps (and they "know" what the discriminant is). Everything here is inside $SL_2(Z)=Aut(Z^2)=Aut(H_1(E_p,Z))$ which acts on the real plane, and up to scalars it acts on the projective real line whose universal covering you already know (yes, the real line). One can lift the relation R=1 to a relation among automorphisms of the real line to get a relation R'=1' where now 1' knows the number of terms appearing on R, that is to say the number of singular fibres, which is the conductor of E in this setting. Then the Szpiro bound can be recovered from the relation R'=1'.

And there you have, a derivative-free proof of the Szpiro conjecture for function fields (a bit shocking at least for me the first time I saw it). All the diophantine information being supplied by fundamental groups.

Are you thinking of "Geometry of algebraic points", Zhang's address to the 1998 International Congress of Chinese Mathematicians, www.math.columbia.edu/~szhang/papers/iccm98.ps ?
–
David SpeyerSep 8 '12 at 14:13